Bias plays an important role in factor analysis and we often implicitly make use of it. The application of non-negativity constraints in alternating least squares (ALS) procedures, for instance, is nothing more than a statement that we are biased toward factor solutions that conform to physical reality. We are willing to accept a model that provides a poorer mathematical fit to the data in return for model that is easier to interpret in physical terms. When components are collinear, a large range of solutions may exist that satisfy the basic constraints and fit the data equally well. In such cases, the introduction of mathematical bias through the application of constraints may select solutions that are less than optimal. Peak shapes may be distorted, for example, or physically discrete chemical phases may appear to be mixed.
The biased ALS (BALS) algorithms of the present invention are designed to offset mathematical bias introduced by constraints in ALS with the goal of achieving factor solutions that are most consistent with physical reality. Methods are also described that are explicitly intended to exploit bias to provide alternative views and provide additional insights into spectral data sets.